hα-Open Sets in Topological Spaces

Abstract

In our work a new type of open sets is introduced and defined as follows: If for each set that is not empty M in 𝑋 , 𝑀 ≠ 𝑋 and 𝑀 ∈ 𝜏 ∝ such that 𝐴 ⊆ int (𝐴 ∪ 𝑀) , then A in (𝑋, 𝜏) is named ℎ ∝ -open set. We also go through the relationship between ℎ ∝ - open sets and a variety of other open set types as h -open sets, open sets, semi-open sets and ∝ -open sets. We proved that each h -open and open set is ℎ ∝ -open and there is no relationship between ∝ -open sets and semi-open sets with ℎ ∝ -open sets. Furthermore, we begin by introducing the concepts of ℎ ∝ -continuous mappings, ℎ ∝ -open mappings, ℎ ∝ -irresolute mappings, and ℎ ∝ -totally continuous mappings, we proved that each h -continuous mapping in any topological space is ℎ ∝ -continuous mapping, each continuous mapping in any topological space is ℎ ∝ -continuous mapping and there is no relationship between ∝ -continuous mappings and semi-continuous mappings with ℎ ∝ -continuous mappings as well as some of its features. Finally, we look at some of the new class's separation axioms.